Method and apparatus for estimating clock deviations, for virtual synchronization of free-running clocks and for determining the position of a movable object

ABSTRACT

In a method for estimating a deviation between a free-running transmitter clock and a reference clock, at a receiver stationary with respect to a transmitter, a transmitter signal generated by the transmitter on the basis of the transmitter clock is received. On the basis of the reference clock, a time of arrival of the transmitter signal and a beat phase of the transmitter signals carrier is determined. On the basis of a clock error model, the time of arrival and the beat phase, the deviation between the transmitter clock and the reference clock is estimated. The clock error model is derived by fitting a correlation function of a stochastic model to a measured auto correlation function of the transmitter clock. Deviations for a plurality of transmitters may be estimated and the transmitters may be virtually synchronized based on the estimations.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority from European Patent Application No.09003962.9, which was filed on Mar. 19, 2009, and is incorporated hereinby reference in its entirety.

BACKGROUND OF THE INVENTION

Embodiments of the invention relate to a method and an apparatus forestimating deviations between free running transmitter clocks and areference clock. In embodiments of the invention, the estimateddeviations are used to virtually synchronize transmitters using thefree-running transmitter clocks. In embodiments of the invention,deviation between a plurality of free-running transmitter clocks and thereference clock is estimated at a stationary receiver, the estimateddeviations are sent to a movable object, wherein the movable object usesthe estimated deviations and transmitter signals received from theplurality of free-running transmitter clocks to determine its position.

Synchronization is a pivotal element in any accurate positioning systembased on time-of-arrival (TOA) or time-difference-of-arrival (TDOA)measurements. There are numerous publications on synchronizationstrategies for local navigation systems. Most of them build uponcomplex, fixed infrastructure or intelligent, expensive transmitters,[1] and [2].

A system comprising low-cost free-running transmitters which arevirtually synchronized by means of a single, fixed reference stationwould be a solution. The reference station would then distribute thesynchronization parameters wirelessly to the rovers. Obviously, such asolution means that any receiver in the system can adequately model thebehavior of the different transmitter clocks. Most published clock errormodels are particularly suitable for long-term stable accurate clocks[3], [4], [5], [6].

All these models identify the types and determine the parameters of thenoise components from plots of the Allan Variance or of the powerspectral densities by a method called power-law noise identificationwhich subdivides the plot into regions of different slopes. Theparameters extracted from the Allan Variance can be used to set up arandom process modeling the clock behavior. A drawback of this method isits limited applicability: some of the noise components are hard tomodel by a rational transfer function. Moreover, all the mentionedstudies focus on the frequency modulate (FM) components: white FM,flicker FM, and random walk FM.

When short-term stability is of interest, phase modulated (PM)components: white and flicker PM noise, if present, are also taken intoaccount. In this case, power-law noise identification by simply readingthe noise parameters from the Allan Variance leads to non rationaltransfer functions.

A different approach to noise identification is based on theautocorrelation function of the phase noise as presented in [7], appliedthere to any type of noise.

US 2006/029009 A1 discloses a system of measuring the range betweennodes in a wireless communications network with one-way data transfer,where each node periodically transmits a message that containsinformation regarding neighboring nodes from which any prior messageshave been received by the transmitting node. A node receives themessages transmitted from neighboring nodes in the network, and recordsthe times of arrival of the received messages. The node receiving thosemessages can thus determine the respective distances between itself andthe neighboring nodes based on the respective time of arrivals of thereceived messages and the respective information included in therespective messages.

Mouly M; Dornstetter J-L: “The Pseudo-Synchronisation, a CostlessFeature to Obtain the Gains of a Synchronised Cellular Network” MRCMobile Radio Conference, XX, XX, 1 Nov. 1991 (Nov. 1, 1991), pages51-55, XP000391318, disclose a pseudo-synchronization scheme for acellular radio telephone system. An accurate knowledge of phasedifferences between the time basis of different base stations ismaintained and each base station stores the time difference with itsneighbors. Once known, the time differences are sent to the entityneeding them at the moment they need them, for instance to a mobilestation when ordered to another cell.

Carpenter R; Lee T: “A stable clock error model using coupled firs andsecond-order gauss-markov processes” Advances in the AstronauticalSciences—Space Flight Mechanics 2008—Advances in the AstronauticalSciences, Proceedings of the AAS/AIAA Space Flight Mechanics Meeting2008 Univelt Inc. US, 31 Dec. 2008 (Dec. 31, 2008), pages 1-13,XP002545930, teaches a stable clock error model using coupled first—andsecond—order gauss-markov processes.

Nicola Altan; Erwin P Rathgeb Ed—David Coudert; David Simplot-RYL, IvanStojomenovic: “Opportunistic Clock Synchronization in a Beacon EnabledWireless Sensor Network” 10 Sep. 2008 (Sep. 10, 2008), AD-HOC, Mobileand Wireless Networks; 20080910 Springer Berlin Heidelberg, Berlin,Heidelberg, page(s) 15-28, XP019102619 ISBN: 9783540852087, teach aclock synchronization in a beacon enabled wireless sensor networkconsisting of a large number of tiny inexpensive sensor nodes. A timesynchronization mechanism based on the usage of a Kalman Filter on asmoothed sequence of measured beacon intervals is proposed. A globalclock synchronization is introduced.

SUMMARY

According to an embodiment, a method for estimating a deviation betweena free-running transmitter clock and a reference clock may have thesteps of: at a receiver stationary with respect to a transmitter,receiving a transmitter signal generated by the transmitter on the basisof the transmitter clock; on the basis of the reference clockdetermining a time of arrival of the transmitter signal and a beat phaseof the transmitter signals carrier; and on the basis of a clock errormodel, the time of arrival and the beat phase, estimating the deviationbetween the transmitter clock and the reference clock, wherein the clockerror model is derived by fitting a correlation function of a stochasticmodel to a measured auto correlation function of the transmitter clock.

According to another embodiment, a method for determining a position ofa movable object may have the steps of: at the movable object, receivingan estimated deviation between each transmitter clock of a plurality oftransmitters and a reference clock of a receiver stationary with respectto the plurality of transmitters and receiving signals generated basedon the free-running transmitter clocks from the plurality oftransmitters, and calculating the position of the movable object using atime of arrival of the received signals from the plurality oftransmitters and transmission times of the received signals corrected bythe estimated deviations.

According to another embodiment, an apparatus for estimating a deviationbetween a free-running transmitter clock and a reference clock may have:a receiver configured to receive a transmitter signal generated by atransmitter stationary with respect to the receiver, on the basis of thetransmitter clock; a processor configured to: on the basis of thereference clock, determine a time of arrival of the transmitter signaland a beat phase of the transmitter signals carrier; and on the basis ofa clock error model, the time of arrival and the beat phase, estimatethe deviation between the transmitter clock and the reference clock,wherein the clock error model is derived by fitting a correlationfunction of a stochastic model to a measured auto correlation functionof the transmitter clock.

According to another embodiment, an apparatus for determining a positionof a movable object may have: a receiver configured to receive estimateddeviations between each transmitter clock of a plurality of transmittersand a reference clock of a receiver stationary with respect to theplurality of transmitters, and to receive signals generated based on thefree-running transmitter clocks from the plurality of transmitters, anda processor configured to calculate the position of the movable objectusing a time of arrival of the received signals from the plurality oftransmitters and transmission times of the received signals corrected bythe estimated deviations.

An embodiment may have a movable object having apparatus for determininga position of a movable object as mentioned above.

Another embodiment may have a system having a plurality of stationarytransmitters spaced from each other, an apparatus for estimating adeviation between a free-running transmitter clock and a reference clockas mentioned above, which is stationary with respect to thetransmitters, and a movable object having apparatus for determining aposition of a movable object as mentioned above.

Another embodiment may have a computer program having program code forperforming, when running on a computer, a method for estimating adeviation between a free-running transmitter clock and a reference clockas mentioned above.

In embodiments of the invention, a stochastic model is used forestimating the clock deviation. The stochastic model is derived from ameasured autocorrelation function. The estimated deviation may bedetermined by a stationary reference unit and may be distributed to amovable object, representing for a receiver in the system. The mobileobject or a plurality of mobile objects (receivers) may use the samemodel to predict or extrapolate current clock deviations betweenupdates.

Embodiments of the invention are based on the recognition that measuredautocorrelation functions for transmitter clocks resemble theautocorrelation function of flicker PM noise and that they are also verysimilar to correlation functions of second order Markov processes.

In embodiments of the invention, the stochastic model is a second orderMarkov process and the correlation function of the stochastic model isthe correlation function of a second order Markov process. Inembodiments of the invention, the deviation is estimated using a Kalmanfilter, wherein coefficients of the system matrix of the Kalman filterare determined based on the clock error model.

In embodiments of the invention, deviations for a plurality oftransmitters are estimated and the estimated deviations for theplurality of transmitters are used to virtually synchronize theplurality of transmitters.

Embodiments of the invention relate to a method for virtualsynchronization of free-running transmitters within a local navigationsystem. The transmitters have free-running oscillators and need nocommunication with other transmitters within the system. They need notbe explicitly connected to a master clock, but a synchronization may beperformed in a virtual manner at a reference station.

In embodiments of the invention, estimated deviations between each of aplurality of free-running transmitter clocks and the reference clock areused to virtual synchronize signals, which are generated by theplurality of transmitters based on the free-running transmitter clocksand which are received at a movable object. In this regard, virtualsynchronization means that respective transmission times of the signals,which are generated and transmitted based on the free-running clocks,are corrected based on the estimated deviations upon receipt of thesesignals. In embodiments of the invention, such signals and the estimateddeviations are used to calculate the position of a movable object.

In embodiments of the invention, the movable object calculatesextrapolated deviations using an estimated or predicted state vector ina state equation of a Kalman filter corresponding to a Kalman filterused in the stationary receiver.

Embodiments of the invention relate to a movable object comprising anapparatus for determining a position of the movable object. Embodimentsof the invention relate to a system comprising a plurality of stationarytransmitters spaced from each other, an apparatus for estimating thedeviation between the free-running transmitter clock and the referenceclock for each of the plurality of stationary transmitters, and amovable object.

According to embodiments of the invention, a stationary referencestation receives the transmitter signals and calculates the clockdeviations using a stochastic model. Subsequently, the reference stationtransmits these deviations to the mobile receivers (rovers) so that theymay correct their data and calculate their positions accordingly. To bemore specific, the deviations may be used to correct the transmissiontimes of the signals, which are used at the movable object incalculating the pseudo-distance between the movable object and thetransmitter. Between updates from the stationary reference station, themobile receiver can predict or extrapolate the clock deviations using anequal model.

Accordingly, embodiments of the invention permit for providing a sharedtime basis for all the components existing within a system so thatposition calculation becomes possible. According to embodiments of theinvention, this is achieved using a stationary reference station, whichsends deviation information associated with a plurality of free-runningtransmitter clocks to a mobile receiver. Thus, the transmitters may befree-running and are not to be synchronized, as in conventional systems,in which explicit synchronization of the transmitters via cables or byusing intelligent transmitters is achieved, i.e. communication takesplace in both directions between the transmitter and the receiver.

Embodiments of the invention are beneficial in that there is nocommunication necessary between the transmitters and the transmittersneed not perform any complicated calculations. Moreover, low-costoscillators may be used in the inventive approach. According toembodiments of the invention, the entire synchronization may beperformed via radio and, therefore, the cost for the infrastructure maybe low.

A communication channel between the reference station and transmittersso as to perform corrections directly at the transmitters, while thereference station is used for monitoring, is not necessary according tothe invention.

In the following specification, navigation within local systems will bedescribed as an application of the invention. It is, however, clear fora person skilled in the art that it is feasible to employ the inventiveapproach for synchronizing other systems, in which receiving units use acommon time base or at least a known time base.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will be described referring to theaccompanying drawings, in which:

FIG. 1 shows a schematic view of a local navigation system, in whichembodiments of the invention may be implemented;

FIG. 2 shows variation of clock deviations calculated as single phasedifferences for six transmitters;

FIG. 3 shows Allan deviation of beat phase measurements for sixfree-running transmitters;

FIG. 4 shows auto correlation of clock variations for six transmitters;

FIG. 5 shows a clock estimation error at a reference receiver;

FIG. 6 shows an impact of an update time interval using estimation onlyat a reference receiver;

FIG. 7 shows the impact of an update time interval using an estimationat a reference receiver and at a movable object;

FIG. 8 shows the influence of a time interval between correction; and

FIG. 9 shows variations of pseudo distance estimates (a) without and (b)with transmitter clock deviation correction.

DETAILED DESCRIPTION OF THE INVENTION

As it is shown in FIG. 1, embodiments of the invention relate to a localmicrowave locating system comprising a couple of asynchronoustransmitters 10, a (unique) receiving reference station 12 and one ormore movable objects 14, such as receiving rovers, wherein the positionsof the transmitters 10 and the reference station 12 are fixed and knownand the position(s) of the movable object(s) are to be determined.

The transmitters 10 may comprise free-running oscillators for providinga free-running transmitter clock and a carrier for transmitter signals30 sent by the transmitters 10, the carrier having a carrier frequencyand a carrier phase. Transmission times of the transmitter signals 30depend on the free-running transmitter clock.

The stationary receiver station 12 comprises a transceiver 20 and aprocessor 22. The movable object 14 comprises a receiver 24 and aprocessor 26.

The transceiver 20 is configured to receive the transmitter signals 30from the plurality of transmitters 10. The transceiver 20 is furtherconfigured to transmit deviation information 32 to the receiver 24 ofthe movable object 14. The processor 22 is configured to derive thedeviation information for each of the transmitters 10 as explained infurther detail below.

The receiver 24 is configured to receive signals 30 from the pluralityof transmitters 10, which are generated based on the free-runningtransmitter clocks. The signals may be identical to the transmittersignals received at the stationary unit 12 and, therefore, are providedwith the same reference number. In addition, the receiver 24 isconfigured to receive the deviation information 32 from the stationaryunit 12. The processor 26 is configured to calculate the position of themobile object 14 using the deviation information 32 and the transmittersignals 30.

The processor 26 may be configured to perform time-of-arrival ortime-difference-of-arrival measurements to determine the distancebetween the movable object 14 and at least some of the transmitters 10in order to calculate the position based thereon.

Generally, the processor 26 may be configured to calculate the distancebetween the movable object 14 and a respective one of the transmittersusing the time of arrival of the signal 30 from the transmitter and thetransmission time. In order to virtually synchronize the free-runningtransmitter, the estimated deviation of the respective transmitter isused to correct the transmission time used to calculate the distance.The transmission time, which is corrected, may be a predefinedtransmission time known to the movable object or may be a transmissiontime forwarded to the movable object along with the transmitter signal.

The calculated distances may then be used to calculate the position ofthe movable object in a known manner.

In the following specification, which relates to an embodiment of theinvention, deriving of a stable model for clock deviations of low-costtransmitters is explained. The following specification is subdividedinto sections I to V.

Section I presents measurements indicating the predominance of noisecomponents which can be modeled more easily by approximating theautocorrelation function rather than using power-law techniques.Applying such a model, Section II introduces a Kalman filter whichestimates the state of a particular transmitter clock. Section IIIpresents the local microwave locating system used as test platform forverifying the algorithms. The use and effectiveness of the clock modelsare discussed in Section IV. Finally, conclusions are drawn in SectionV.

I. STOCHASTIC NATURE OF CLOCK DEVIATIONS

Each transmitter and receiver in the navigation system generates a timebase using its own oscillator. Because the oscillators are not ideal, itis natural that some deviations between the time bases of differentcomponents appear. These deviations affect the measurements in thenavigation system. In this section the relationship between theobservables and the clock deviations is derived, the model used fordescribing the clock deviations is presented. A particular transmitter iuses its own time base to determine the exact time point t_(i) to send asignal. This signal travels a distance d_(ri) and arrives at thereceiver r at time t_(ri), according to the receiver time scale. Thetransmitter clock deviation δt_(i)[t] is defined as the differencebetween its time base t_(i)[t] and a reference time t, which is in thiscase the receiver time base.δt _(i) [t]=t−t _(i) [t]  (1)

Time of arrival measurements at the receiver take the form

$\begin{matrix}{t_{ri} = {t_{i} + {\delta\;{t_{i}\left\lbrack t_{ri} \right\rbrack}} + \frac{d_{ri}}{c}}} & (2)\end{matrix}$where d_(ri) corresponds to the distance between transmitter andreceiver and c is the speed of light. It was implicitly assumed that thetransmitter clock deviation δt_(i)[t] does not change noticeably duringthe brief period of transmission.

Besides time of arrival (TOA) the receivers in the system measure thecarrier beat phase φ_(bi), i.e. the difference between the receiverphase φ_(r) at reception time t_(ri) and transmitter phase φ_(i) attransmission time t^(s) using the receiver time as reference:φ_(bi)=φ_(r) [t _(ri)]−φ_(i) [t ^(s)]  (3)

Both, receiver and transmitter generate the carrier signal with anominal frequency f₀, which gives the relationship between transmittertime t_(i)[t] and phase φ_(i)[t]:φ_(i) [t ^(s)]=2πf ₀ t _(i) [t ^(s)]  (4)

Variations in the frequency f_(i)[t] are the prime reason for the clockand phase deviations of (1) and (3). An ideal behavior of the receiverclocks was assumed, since the transmitter clocks are much cheaper. Using(1) to replace t_(i)[t^(s)] in the transmitter phase φ_(i), we obtain:

$\begin{matrix}{{2\pi{\int_{0}^{t^{s}}{{f_{i}\lbrack t\rbrack}{\mathbb{d}t}}}} = {{\phi_{i}\left\lbrack t^{s} \right\rbrack} = {2\pi\;{f_{0}\left( {t^{s} - {\delta\;{t_{i}\left\lbrack t^{s} \right\rbrack}}} \right)}}}} & (5)\end{matrix}$

The receiver phase φ_(r)[t_(ri)] takes the form

$\begin{matrix}{{\phi_{r}\left\lbrack t_{ri} \right\rbrack} = {{2\pi\; f_{0}t_{ri}} = {2\pi\;{f_{0}\left( {t^{s} + \frac{d_{ri}}{c}} \right)}}}} & (6)\end{matrix}$replacing time of reception t_(ri) by time of transmission t^(s) plustravel time d_(ri)/c. Inserting (5) and (6) into (3) the beat phasetakes the form

$\begin{matrix}{{\phi_{bi}\left\lbrack t_{ri} \right\rbrack} = {{2\pi\; f_{0}\delta\;{t_{i}\left\lbrack t^{s} \right\rbrack}} + {2\pi\; f_{0}\frac{d_{ri}}{c}}}} & (7)\end{matrix}$

The variation of the beat phase corresponds to the variation of thetransmitter clock if the distance d_(ri) between transmitter andreceiver remains constant and the speed of light does not vary.Differentiation with respect to time then yields (assuming againδt_(i)[t^(s)]=δt_(i)[t_(ri)], i.e. slowly changing clocks):

$\begin{matrix}{\frac{\partial\phi_{bi}}{\partial t} = {2\pi\; f_{0}\frac{{\partial\delta}\; t_{i}}{\partial t}}} & (8)\end{matrix}$

The assumption of constant distance holds true for the referencereceiver and the transmitters since they are static.

The relationships shown in (2), (7) and (8) demonstrate that both, TOAand phase measurements on a static receiver provide information aboutthe behavior of the transmitter clocks. Such measurements will be usedin Section II to determine clock deviations.

In (9) below, based on single phase differences an approximation of thederivative in (8) is given. The upper index k indicates that the beatphase is sampled at discrete time points.

$\begin{matrix}{{\frac{\partial\phi_{bi}}{\partial t} \approx \frac{\Delta\phi}{\Delta\; t}} = {\frac{\phi_{bi}^{k} - \phi_{bi}^{k - 1}}{t_{ri}^{k} - t_{ri}^{k - 1}}.}} & (9)\end{matrix}$

For six transmitters Tx1 to Tx6 of the microwave locating system, FIG. 2shows the variation of the clock error calculated according to (8).

For convenience, the values are expressed as speed in m/s, as quitecommon in positioning systems like GPS. FIG. 2 clearly shows thenon-deterministic nature of clock errors. Traditional techniques fortime domain stability analysis of clocks use the Allan Variance as ameasure of clock stability.

By the Allan Variance it is possible to identify the type of noisepresent in the clock errors (power-law noise identification). Beforeciting the definition of the Allan Variance the fractional frequencyerror y(t) and the error phase δφ are defined. Fractional frequencyerror is the ratio of the frequency error of and the nominal frequencyf₀. The error phase is the error in the phase produced by the frequencyvariations. The relationship between y(t) and δφ is:

$\begin{matrix}{{y(t)} = {\frac{\delta\; f}{f_{0}} = {\frac{1}{2\pi\; f_{0}}\frac{\partial{\delta\phi}}{\partial t}}}} & (10)\end{matrix}$

The Allan Variance is calculated as the average of the squareddifference between two samples of the fractional frequency y(t) taken ata distance τ, where τ is the time interval between the samples i andi+τ.

$\begin{matrix}{{\sigma_{y}^{2}(\tau)} = {\frac{1}{2\left( {M - 1} \right)}{\sum\limits_{i = 1}^{M - 1}\left( {{y\left( t_{i + 1} \right)} - {y\left( t_{i} \right)}} \right)^{2}}}} & (11)\end{matrix}$

Allan Deviation is the square root of the Allan Variance. In setupvariations of the phase error δφ correspond to variations of the beatphase φ_(bi) provided speed of light and distance between transmitterand (reference) receiver do not change, cf. (7). The Allan Deviation isused to identify relevant noise components of the transmitter clocks. InFIG. 3 the Allan Deviation of the signals presented before in FIG. 2 isshown. For time intervals smaller than 0.1 seconds (left of verticalline in FIG. 3) the slope of the Allan Deviation is approximately minusone. This demonstrates the predominance of flicker PM noise. The AllanVariance is closely related to the power spectral density of thefractional frequency error y(t). The slope of flicker PM noise in apower density plot S_(y)(f) is plus one, i.e. S_(y)(f) αf [8]. In thiscase it is hard to model the noise component by applying a filter withrational transfer function to white noise.

Instead of deriving the clock error model directly from the AllanDeviation, according to the invention, it is proposed to derive it byfitting the correlation function of the stochastic model, such as thenoise model, to measured autocorrelation functions.

In FIG. 4 measured autocorrelation functions for the six transmitterclocks Tx1 to Tx6 are shown. They resemble the autocorrelation functionof flicker PM noise given in Table 2 of [7]. They are also very similarto correlation functions of second order Markov processes. Second orderMarkov processes are characterized by an attenuation factor ζ and anatural frequency ω_(n). Their transfer function and the correspondingcorrelation function are given in [9]:

$\begin{matrix}{{H(s)} = {\frac{{as} + b}{s^{2} + {2{\zeta\omega}_{n}s} + \omega_{n}^{2}}.}} & (12) \\{{R(\tau)} = {\frac{\sigma^{2}}{\cos\;\theta}{\mathbb{e}}^{{\zeta\omega}_{n}{\tau }}{\cos\left\lbrack {{\sqrt{1 - \zeta^{2}}\omega_{n}{\tau }} - \theta} \right\rbrack}}} & (13)\end{matrix}$

The parameters σ and θ are constants. Taking a=0 and fitting theautocorrelation function of the Markov model to the functions presentedin FIG. 4, values for the parameters ζ and ω_(n) were obtained. Theparameters ζ and ω_(n) determine on the one hand the frequency of theoscillations and the size of the secondary peaks in the correlationfunctions and on the other hand the poles of the transfer function,which define the form of power spectral density.

In FIG. 3 the Allan Deviation of simulated data is shown (solid line)for a second order Markov process modeling the clock deviations as justdescribed. Clearly, the simulated clock deviations can model properlythe behavior observed in the real data including the flicker PM noisecomponent.

II. KALMAN FILTER

This section describes the state variable representation of the modelpresented in the Section I and shows the development of the KalmanFilter equations. In general, the state space representation of a linearsystem used in a Kalman filter is described by a state equation (14) andan observation equation (15).{dot over (X)}(t)=FX(t)+W(t)  (14)Z(t)=HX(t)+V(t)  (15)X represents the state vector, F the system matrix, W process noise, Zthe measurement vector, H the output matrix and V the measurement noise.

For the clock model the absolute clock error δt_(i), its derivativeδ{dot over (t)}_(i) and the second derivative δ{umlaut over (t)}_(i)were chosen as state variables.X _(i) =[δt _(i) δ{dot over (t)} _(i) δ{umlaut over (t)} _(i)]^(T)  (16)

The system matrix F takes the form in (17). The last two lines in thismatrix correspond to the second order Markov model for the variation ofthe clock deviation, as described in Section I.

$\begin{matrix}{F = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & {- \omega_{n}^{2}} & {- {\zeta\omega}_{n}}\end{bmatrix}} & (17)\end{matrix}$

Measurements of the absolute clock deviation δt_(i) were calculatedusing (2). The transmitter is programmed to send brief sequences with aperiod T_(i). Therefore, the transmit time at the transmitter t_(i) canbe replaced by K·T_(i), where the integer K is a counter indicating thenumber of sent sequences. Single phase differences as calculated in (9)were taken as measurements of the clock error variation. The measurementvector and the corresponding output matrix take the form:

$\begin{matrix}{{Z_{i} = \begin{bmatrix}{t_{ri} - {K \cdot T_{i}} - \frac{d_{ri}}{c}} \\{{{\Delta\phi}/\Delta}\; t}\end{bmatrix}}{H = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\end{bmatrix}}} & (18)\end{matrix}$

The clock state of each transmitter is estimated in the referencereceiver by separate Kalman filters and transmitted to the rover. Therover uses Kalman predictors to estimate the clock deviations. The modelfor the predictors comprises only the state equation (14). Each time anew clock state estimate arrives at the rover the state vector of thecorresponding transmitter is updated. With the estimated clockdeviations the pseudo distances between transmitters and rover arecalculated. These are used to estimate the rover position and the clockdeviations between reference station and rover.

III. TEST PLATFORM

The test platform used to generate data in order to verify thealgorithms comprises a simulation tool and a real locating system. Thesimulation tool was developed in Matlab. It allows generating controlledmeasurements in order to proof the efficiency of the algorithms.Furthermore, the clock estimation filters for both reference station androver as well as the position calculation were developed in Matlab. Theyare able to process both simulated data and data from the real system.The real system consists of a set of asynchronous free runningtransmitters and receivers located at known positions.

Transmitters are in charge of the generation and periodic transmissionof brief code sequences in the ISM band at 2.4 GHz. Each transmitter isprogrammed to emit brief sequences with a period of approximately 5 ms.Twelve static receivers and six transmitters were installed around afootball field. The clocks on the transmitters are low cost crystaloscillators. For all tests presented here, the receivers aresynchronized by a fiber optic network to a master clock, which is basedon an oven-controlled crystal oscillators (OCXO).

IV. EXPERIMENTAL RESULTS

The first goal of the experiments is to demonstrate the efficiency ofthe algorithms to estimate clock deviations δt_(i). A set ofmeasurements (TOAs and beat phases) for six free running transmittersand two receivers (reference station and rover) were generated using thesimulation tool. The simulated reference station data are used toestimate the transmitter clock deviations. Because of simulated data, itis possible to calculate the error in the estimation. In FIG. 5 theerror of one transmitter clock deviation estimation versus time at thereference station is shown. The standard deviation of measurement noiseused for these simulations was 150 ps for the time measurements and 0.04rad for the phase measurements. The noise in the clock deviationestimation is zero mean, its deviation is about 25 ps.

The synchronization strategy necessitates the reference station, whichestimates the clock deviations δt_(i), to process data at the same rateas signals are received. The estimated clock deviations should betransmitted at a lower rate in order to keep the data traffic betweenreference station and rovers at appropriate levels. Therefore, it isimportant to analyze the impact of the clock parameter update rate onthe accuracy of the position estimates. The rovers must be able topredict the clock deviation between updates.

A first approach is a zero order prediction, i.e. the last clockdeviation estimate received from the reference station is used until anew correction arrives at the rover. For these experiments update timeintervals of 0.005, 1 and 5 seconds were chosen. FIG. 6 shows theresults obtained by following this approach. The larger the updateinterval, the faster the errors grow up to unacceptable levels above thenanosecond range. Hence this approach is impractical for positioningwith sub-meter level accuracy.

Another approach uses Kalman predictors at the rovers to extrapolate thetransmitter clock deviations between updates received from the referencestation. Both, reference station and rover use the same Kalman filter asexplained at the end of Section II. Obviously the efficiency of theprediction is affected by the update rate of the correction values. Thisis underlined by FIG. 7, which shows clock deviation estimation errorsat the rover for the three update time intervals 0.005, 1 and 5 seconds.The errors stay in a sub-nanosecond range. The superiority of theproposed approach to use Kalman predictors at the rovers is obvious.

In FIG. 8 the effect on the positioning accuracy versus time is shown.From top to bottom the update time interval has been increased. Eachsubplot shows the positioning accuracy for the three coordinates X, Y,and Z. For time intervals up to one second, the estimation works inacceptable limits and there is no significant change in accuracy. Thenoise observed in the position is due to noise in the TOA-measurements,which have not been filtered in this experiment. For longer updateintervals the error increases noticeably and peaks just before anupdate. It can be observed that the scaling of the vertical axis hasbeen changed for each update interval.

Real data have been collected during one hour from the test system tocheck the algorithms. One of the static receivers was taken as referencestation to estimate the transmitter clock model states as presented inSection II. A second receiver used the estimated states from thereference station to estimate transmitter clock deviations δt_(i) like arover. The rover TOA measurements t_(ri) were corrected using thepredicted transmitter clock deviations. Pseudo distances can becalculated by subtracting time of transmission estimates K·T_(i) fromthe rover TOA measurements.

FIG. 9( a) shows how the pseudo distance estimates diverge with time ifthe transmitter clock deviations are not modeled, i.e. the rover TOAmeasurements are not corrected to take into account the transmitterclock deviation. After one hour the pseudo distance estimates havedecreased by 4·10⁶ m though the distance between rover and transmitterdid not change. FIG. 9( b) shows the corresponding result if the roverTOA measurements are corrected. The algorithm is able to compensate thetransmitter clock errors, the pseudo distance estimates vary only in thedecimeter range. It is clear that without the correction of thetransmitter clock deviation the calculation of a reasonable positionestimate by the rover would be impossible.

V. CONCLUSION

For a microwave locating system a method has been proposed tosynchronize virtually all transmitters and receivers in the systemwithout explicitly coupling them to a master clock. To this end, astochastic model for accurately estimating clock deviations has beendevised. The modeling starts from measured autocorrelation functionsinstead of exploiting traditional power-law measures for noiseidentification as, e.g., the Allan Deviation. The autocorrelation methodallows the design of stable filters. This model is able to reflect thebehavior of low cost oscillators characterized by flicker PM noise andother noise components. The stochastic model may be exploited in bothreference receiver and rovers. In the reference receiver, a set ofKalman filters, one for each transmitter, estimates the transmitterclock deviations using TOA and beat phase measurements. In the rover,Kalman predictors forecast the clock deviations between correctionupdates. By simulations based on measured data, the superiority of theKalman predictor compared to a zero order predictor was demonstrated.Furthermore, this approach has been shown to be adequate to provideaccurate position estimates in the submeter range as long as the updatetime interval stays below one second. The algorithms were tested alsowith real data collected in the microwave locating system. Theeffectiveness to compensate transmitter clock deviations wasdemonstrated. Without such a strategy it would be impossible tocalculate reasonable position estimates.

Accordingly, embodiments of the invention relate to a virtualsynchronization of low-cost free-running transmitter clocks in amicrowave locating system. The synchronization strategy may be based onmodeling the transmitter clocks. A stable stochastic model may bedesigned by analyzing the correlation function of the observables in thelocating system, i.e. time of arrival and carrier phase. In embodimentsof the invention, this model is exploited in a Kalman filter and areference receiver estimates the current clock states and distributesthem periodically to the mobile receivers enabling them to predict thetransmitter clock deviations between updates. Referring to FIGS. 6 to 9,the impact of such a synchronization scheme on the position accuracyusing both simulated and real data was analyzed.

Depending on certain implementation requirements, embodiments of theinvention can be implemented in hardware or in software. Theimplementation can be performed using a digital storage medium, inparticular a disk, DVD or a CD having electronically readable controlsignals stored thereon, which cooperate with a programmable computersystem such that the inventive methods are performed. Generally,embodiments of the invention may be a computer program product with aprogram code stored on a machine readable carrier, the program codebeing operative for performing the inventive methods when the computerprogram product runs on a computer. In other words, embodiments of theinventive methods may be implemented as a computer program having aprogram code for performing at least one of the inventive methods whenthe computer program runs on a computer.

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which fall withinthe scope of this invention. It should also be noted that there are manyalternative ways of implementing the methods and compositions of thepresent invention. It is therefore intended that the following appendedclaims be interpreted as including all such alterations, permutations,and equivalents as fall within the true spirit and scope of the presentinvention.

REFERENCES

-   [1] D. Yun and C. Kee, “Centimeter accuracy stand-alone indoor    navigation system by synchronized pseudolite constellation,”    Proceedings of the 15^(th) International Technical Meeting of the    Satellite Division of the Institute of Navigation ION GPS, pp.    213-225, September 2002.-   [2] J. Barnes, C. Rizos, J. Wang, D. Small, G. Voigt, and N.    Gambale, “High Precision Indoor and Outdoor Positioning using    LocataNet,” Journal of Global Positioning Systems, vol. 2, no. 2,    pp. 73-82, 2004.-   [3] L. Galleani, L. Sacerdote, P. Tavella, and C. Zucca, “A    mathematical model for the atomic clock error,” METROLOGIABERLIN-,    vol. 40, no. 3, pp. 257-264, 2003.-   [4] J. Wright, “GPS Composite Clock Analysis,” Time, vol. 8, no.    6, p. 4.-   [5] F. Gonzalez and P. Waller, “Short term GNSS clock    characterization using One-Way carrier phase,” pp. 517-522, 2007.-   [6] P. Daly and I. Kitching, “Characterization of NAVSTAR GPS and    GLONASS on-board clocks,” Aerospace and Electronic Systems Magazine,    IEEE, vol. 5, no. 7, pp. 3-9, 1990.-   [7] W. Riley and C. Greenhall, “Power Law Noise Identification Using    the Lag 1 Autocorrelation,” in Proc. 18th European Frequency and    Time Forum (Guildford, UK), 2004.-   [8] T. Parker, “Characteristics and Sources of Phase Noise in Stable    Oscillators,” pp. 99-110, 1987.-   [9] M. Grewal and A. Andrews, Kalman Filtering: Theory and Practice    Using MATLAB. Wiley, 2001.

1. A method for estimating a deviation between a free-runningtransmitter clock and a reference clock, the method comprising: at areceiver stationary with respect to a transmitter, receiving atransmitter signal generated by the transmitter on the basis of thetransmitter clock; on the basis of the reference clock determining atime of arrival of the transmitter signal and a beat phase of thetransmitter signals carrier; and on the basis of a clock error model,the time of arrival and the beat phase, estimating the deviation betweenthe transmitter clock and the reference clock, wherein the clock errormodel is derived by fitting a correlation function of a stochastic modelto a measured auto correlation function of the transmitter clock andwherein the correlation function of the stochastic model is thecorrelation function of a second order Markov process.
 2. The method ofclaim 1, wherein estimating the deviation comprises: using a Kalmanfilter, wherein coefficients of the system matrix of the Kalman filterare determined based on the clock error model.
 3. The method of claim 2,wherein the deviation between the free-running transmitter clock and thereference clock is output as an estimated state vector of the Kalmanfilter.
 4. The method of claim 1, comprising: estimating a deviationbetween each of a plurality of free-running transmitter clocks and thereference clock, wherein each of the free-running transmitter clocks isassociated with one of a plurality of stationary transmitters spacedfrom each other; and virtually synchronizing the plurality oftransmitters using the estimated deviations.
 5. A method for determininga position of a movable object, comprising: at the movable object,receiving an estimated deviation between each transmitter clock of aplurality of transmitters and a reference clock of a receiver stationarywith respect to the plurality of transmitters and receiving signalsgenerated based on the free-running transmitter clocks from theplurality of transmitters, and calculating the position of the movableobject using a time of arrival of the received signals from theplurality of transmitters and transmission times of the received signalscorrected by the estimated deviations, wherein the deviations areestimated state vectors of a Kalman filter used in the stationaryreceiver, and wherein the movable object calculates extrapolateddeviations using the estimated state vectors in a state equation of aKalman filter corresponding to the Kalman filter used in the stationaryreceiver, and wherein the movable object calculates the extrapolateddeviations in a time period between subsequent estimated state vectorsreceived from the stationary receiver.
 6. An apparatus for estimating adeviation between a free-running transmitter clock and a referenceclock, comprising: a receiver configured to receive a transmitter signalgenerated by a transmitter stationary with respect to the receiver, onthe basis of the transmitter clock; a processor configured to: on thebasis of the reference clock, determine a time of arrival of thetransmitter signal and a beat phase of the transmitter signals carrier;and on the basis of a clock error model, the time of arrival and thebeat phase, estimate the deviation between the transmitter clock and thereference clock, wherein the clock error model is derived by fitting acorrelation function of a stochastic model to a measured autocorrelation function of the transmitter clock and wherein thecorrelation function of the stochastic model is the correlation functionof a second order Markov process.
 7. An apparatus for determining aposition of a movable object, comprising: a receiver configured toreceive estimated deviations between each transmitter clock of aplurality of transmitters and a reference clock of a receiver stationarywith respect to the plurality of transmitters, and to receive signalsgenerated based on the free-running transmitter clocks from theplurality of transmitters, and a processor configured to calculate theposition of the movable object using a time of arrival of the receivedsignals from the plurality of transmitters and transmission times of thereceived signals corrected by the estimated deviations, wherein thedeviations are estimated state vectors of a Kalman filter used in thestationary receiver, and wherein the processor is configured tocalculate extrapolated deviations using the estimated state vectors in astate equation of a Kalman filter corresponding to the Kalman filterused in the stationary receiver, and wherein the processor is configuredto calculate the extrapolated deviations in a time period betweensubsequent estimated state vectors received from the stationaryreceiver.
 8. A movable object comprising an apparatus for determining aposition of a movable object, comprising: a receiver configured toreceive estimated deviations between each transmitter clock of aplurality of transmitters and a reference clock of a receiver stationarywith respect to the plurality of transmitters, and to receive signalsgenerated based on the free-running transmitter clocks from theplurality of transmitters, and a processor configured to calculate theposition of the movable object using a time of arrival of the receivedsignals from the plurality of transmitters and transmission times of thereceived signals corrected by the estimated deviations, wherein thedeviations are estimated state vectors of a Kalman filter used in thestationary receiver, and wherein the processor is configured tocalculate extrapolated deviations using the estimated state vectors in astate equation of a Kalman filter corresponding to the Kalman filterused in the stationary receiver, and wherein the processor is configuredto calculate the extrapolated deviations in a time period betweensubsequent estimated state vectors received from the stationaryreceiver.
 9. A system comprising a plurality of stationary transmittersspaced from each other, an apparatus for estimating a deviation betweena free-running transmitter clock and a reference clock, comprising: areceiver configured to receive a transmitter signal generated by atransmitter stationary with respect to the receiver, on the basis of thetransmitter clock; a processor configured to: on the basis of thereference clock, determine a time of arrival of the transmitter signaland a beat phase of the transmitter signals carrier; and on the basis ofa clock error model, the time of arrival and the beat phase, estimatethe deviation between the transmitter clock and the reference clock,wherein the clock error model is derived by fitting a correlationfunction of a stochastic model of a second order Markov process to ameasured auto correlation function of the transmitter clock, which isstationary with respect to the transmitters, and a movable objectcomprising an apparatus for determining a position of a movable object,comprising: a receiver configured to receive estimated deviationsbetween each transmitter clock of a plurality of transmitters and areference clock of a receiver stationary with respect to the pluralityof transmitters, and to receive signals generated based on thefree-running transmitter clocks from the plurality of transmitters, anda processor configured to calculate the position of the movable objectusing a time of arrival of the received signals from the plurality oftransmitters and transmission times of the received signals corrected bythe estimated deviations.
 10. A non-transitory computer-readable mediumhaving stored thereon a computer program comprising program code forperforming, when running on a computer, a method for estimating adeviation between a free-running transmitter clock and a referenceclock, the method comprising: at a receiver stationary with respect to atransmitter, receiving a transmitter signal generated by the transmitteron the basis of the transmitter clock; on the basis of the referenceclock determining a time of arrival of the transmitter signal and a beatphase of the transmitter signals carrier; and on the basis of a clockerror model, the time of arrival and the beat phase, estimating thedeviation between the transmitter clock and the reference clock, whereinthe clock error model is derived by fitting a correlation function of astochastic model to a measured auto correlation function of thetransmitter clock and wherein the correlation function of the stochasticmodel is the correlation function of a second order Markov process.